DOCSLIB.ORG

Study on Delaunay Tessellations of 1-Irregular Cuboids for 3D Mixed Element Meshes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Study on Delaunay Tessellations of 1-Irregular Cuboids for 3D Mixed Element Meshes Study on Delaunay tessellations of 1-irregular cuboids for 3D mixed element meshes David Contreras and Nancy Hitschfeld-Kahler Department of Computer Science, FCFM, University of Chile, Chile E-mails: dcontrer,[email protected] Abstract Mixed elements meshes based on the modified octree approach con- tain several co-spherical point configurations. While generating Delaunay tessellations to be used together with the finite volume method, it is not necessary to partition them into tetrahedra; co-spherical elements can be used as final elements. This paper presents a study of all co-spherical elements that appear while tessellating a 1-irregular cuboid (cuboid with at most one Steiner point on its edges) with different aspect ratio. Steiner points can be located at any position between the edge endpoints. When Steiner points are located at edge midpoints, 24 co-spherical elements appear while tessellating 1-irregular cubes. By inserting internal faces and edges to these new elements, this numberp is reduced to 13. When 1-irregular cuboids with aspect ratio equal to 2 are tessellated, 10 co- spherical elementsp are required. If 1-irregular cuboids have aspect ratio between 1 and 2, all the tessellations are adequate for the finite volume method. When Steiner points are located at any position, the study was done for a specific Steiner point distribution on a cube. 38 co-spherical elements were required to tessellate all the generated 1-irregular cubes. Statistics about the impact of each new element in the tessellations of 1-irregular cuboids are also included. This study was done by develop- ing an algorithm that construct Delaunay tessellations by starting from a Delaunay tetrahedral mesh built by Qhull. arXiv:1312.1181v1 [cs.CG] 4 Dec 2013 1 Introduction Scientific and engineering problems are usually modeled by a set of partial dif- ferential equations and the solution to these partial differential equations is calculated through the use of numerical methods. In order to get good results, the object being modeled (domain) must be discretized in a proper way respect- ing the requirements imposed by the used numerical method. The discretization (mesh) is usually composed of simple cells (basic elements) that must represent the domain in the best possible way. In particular, we are interested in meshes for the finite volume method [1] which are formed by polygons (in a 2D domain) 1 Study on Delaunay tessellations 2 Figure 1: The seven final elements of the Ω Mesh Generator: (a) Cuboid, (b) Triangular Prism, (c) Quadrilateral Pyramid, (d) Tetrahedron, (e) Tetrahedron Complement, (f) Deformed Prism, and (g) Deformed Tetrahedron Complement. or polyhedra (in a 3D domain), that satisfy the Delaunay condition: the cir- cumcircle in 2D, or circumsphere in 3D, of each element does not contain any other mesh point in its interior [2]. The Delaunay condition is required because we use its dual structure, the Voronoi diagram, to model the control volumes in order to compute an approximated solutions. The basic elements used so far are triangles and quadrilaterals in 2D, and tetrahedra, cuboids, prisms and pyramids in 3D. Meshes composed of different elements types are called mixed element meshes [3]. Mixed element meshes are built on 2D or 3D domains described by sets of points, polygons or polyhedra depending on the application. We have developed a mixed element mesh generator [4] based on an extension of octrees [5, 6]. Our approach starts enclosing the domain in the smallest bounding box (cuboid). Second, this cuboid is continuously refined, at any edge position, by using the geometry information of the domain. That is why this refinement is called inter- section based approach. Once this step finishes, an initial non-conforming mesh composed of tetrahedra, pyramids, prisms, and cuboids is generated that fits the domain geometry. Third, these elements are further refined by bisection, as far as possible, until the density requirements are fulfilled. Fourth, the mesh is done 1-irregular by allowing only one Steiner point on each edge. The current solution is based on patterns but only the most frequently used patterns are available. Then, if a pattern is not available or the element can not be properly tessellated for the finite volume method, new Steiner points are inserted until all 1-irregular elements can be properly tessellated. The current set of seven final elements is shown in Figure 1. The advantage of using a mixed mesh in comparison with a tetrahedral mesh is that the use of different element types reduce the amount of edges, faces and elements in the final mesh. For example, we do not need to divide a cuboid into tetrahedra. On the other hand, a disadvantage is that the equations must be Study on Delaunay tessellations 3 Figure 2: (a) Mixed mesh of a 1-irregular cuboid that satisfies the Delaunay condition, (b) the same mixed mesh and its associated Voronoi diagram. discretized using different elements. Octree based approaches naturally produces co-spherical point sets. A mixed mesh satisfying the Delaunay condition can include all produced co-spherical elements as shown in Figure 2. The final elements in this example are five pyra- mids and four tetrahedra. The goal of this paper is to study the co-spherical elements that can appear while tessellating 1-irregular cuboids generated by using a bisection and inter- section based approach and to analyze how useful would be to include the new elements in the final element set. In particular, this paper gives the number and shape of new co-spherical elements needed to tessellate (a) all 1-irregular cuboids generated by the bisection approach and (b) some particular 1-irregular cuboids generated by the intersection refinement approach. In addition, statis- tics associated with particular tessellations are presented such as the frequency each co-spherical element is used and the number of tessellations that can be used with the finite volume method without adding extra vertices. The analy- sis of the 1-irregular cuboid tessellations was done under different criteria that affect the amount of generated co-spherical elements. We have focused this work on the analysis of the tessellations of 1-irregular cuboids because this element is the one that more frequently appears when meshes are generated by a modified octree approach. A theoretical study on the number of different 1-irregular cuboid configurations that can appear either by using a bisection or an intersection based approach was published in [7]. We use the results of that work as starting point for this study. This paper is organized as follows: Section 2 describes the bisection and intersec- tion refinement approaches. Section 3 presents briefly the developed algorithm to compute Delaunay tessellations. Section 4 and Section 5 give the results obtained by applying the algorithm to 1-irregular cuboids generated by a bisec- tion and an intersection based approach, respectively. Section 6 includes our conclusions. Study on Delaunay tessellations 4 Figure 3: Cuboid and its splits into two, four and eight cuboids using a bisection based approach. Figure 4: The bisection-refined cuboid at the left produces an 1-irregular ele- ment like the cuboid at the right . 2 Basic concepts This section describes some ideas in order to understand how 1-irregular cuboids are generated. 2.1 Bisection based approach 1-irregular configurations In this approach, the Steiner points inserted at the refinement phase are always located at the edge midpoints. Cuboids can be refined into two, four or eight smaller cuboids as shown in Figure 3. This refinement produces neighboring cuboids with Steiner points located at the edge midpoints. Those 1-irregular cuboids are larger than the already refined neighbor cuboid as shown in Figure 4. 2.2 Intersection based approach 1-irregular configurations While using an intersection based approach, the Steiner points are not neces- sarily located at the edge midpoints. In general, there are no constrains on the location of the Steiner points, except by the fact that parallel edges must be divided in the same relative position to ensure the generation of cuboids and not any other polyhedron. Figure 5 shows an example of this approach. This refinement produces neighboring cuboids with Steiner points located at any edge position. Those 1-irregular cuboids are larger than the already refined neighbor cuboid as shown in Figure 4. Study on Delaunay tessellations 5 Figure 5: Cuboid and its splits into two, four and eight cuboids using an inter- section based approach. Figure 6: The intersection-refined cuboid to the left produces an 1-irregular element like the cuboid to the right. 3 Algorithm In order to count the number of new co-spherical elements than can appear and to recognize their shape, we have developed an algorithm that executes the followings steps: 1. Build the point configuration of a 1-irregular cuboid by specifying the coordinates of the cuboid vertices and its Steiner points. 2. Build a Delaunay tetrahedral mesh for this point configuration by using QHull [8]1. 3. Join tetrahedra to form the largest possible co-spherical elements. 4. Identify each final co-spherical polyhedron. Qhull divides co-spherical point configurations into a set of tetrahedra by adding an artificial point that is not part of the input. Then, we use this fact to recognize the faces that form a co-spherical polyhedron and later to recognize which element is. 4 Results: Bisection based approach This section describes the results obtained by applying the previous algorithm to the 4096 (212) 1-irregular configurations that can be generated using a bisection based approach. First, the new co-spherical elements are shown. Then, their impact in all the tessellations is analyzed and finally, the tessellations that can be used with the finite volume method are characterized.
Load more

Recommended publications
  • Unit 6 Visualising Solid Shapes(Final)
    • 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure.
    [Show full text]
  • Uniform Panoploid Tetracombs
    Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs.
    [Show full text]
  • Hexagonal Antiprism Tetragonal Bipyramid Dodecahedron
    Call List hexagonal antiprism tetragonal bipyramid dodecahedron hemisphere icosahedron cube triangular bipyramid sphere octahedron cone triangular prism pentagonal bipyramid torus cylinder square­based pyramid octagonal prism cuboid hexagonal prism pentagonal prism tetrahedron cube octahedron square antiprism ellipsoid pentagonal antiprism spheroid Created using www.BingoCardPrinter.com B I N G O hexagonal triangular square­based tetrahedron antiprism cube prism pyramid tetragonal triangular pentagonal octagonal cube bipyramid bipyramid bipyramid prism octahedron Free square dodecahedron sphere Space cuboid antiprism hexagonal hemisphere octahedron torus prism ellipsoid pentagonal pentagonal icosahedron cone cylinder prism antiprism Created using www.BingoCardPrinter.com B I N G O triangular pentagonal triangular hemisphere cube prism antiprism bipyramid pentagonal hexagonal tetragonal torus bipyramid prism bipyramid cone square Free hexagonal octagonal tetrahedron antiprism Space antiprism prism square­based dodecahedron ellipsoid cylinder octahedron pyramid pentagonal icosahedron sphere prism cuboid spheroid Created using www.BingoCardPrinter.com B I N G O cube hexagonal triangular icosahedron octahedron prism torus prism octagonal square dodecahedron hemisphere spheroid prism antiprism Free pentagonal octahedron square­based pyramid Space cube antiprism hexagonal pentagonal triangular cone antiprism cuboid bipyramid bipyramid tetragonal cylinder tetrahedron ellipsoid bipyramid sphere Created using www.BingoCardPrinter.com B I N G O
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Step 1: 3D Shapes
    Year 1 – Autumn Block 3 – Shape Step 1: 3D Shapes © Classroom Secrets Limited 2018 Introduction Match the shape to its correct name. A. B. C. D. E. cylinder square- cuboid cube sphere based pyramid © Classroom Secrets Limited 2018 Introduction Match the shape to its correct name. A. B. C. D. E. cube sphere square- cylinder cuboid based pyramid © Classroom Secrets Limited 2018 Varied Fluency 1 True or false? The shape below is a cone. © Classroom Secrets Limited 2018 Varied Fluency 1 True or false? The shape below is a cone. False, the shape is a triangular-based pyramid. © Classroom Secrets Limited 2018 Varied Fluency 2 Circle the correct name of the shape below. cylinder pyramid cuboid © Classroom Secrets Limited 2018 Varied Fluency 2 Circle the correct name of the shape below. cylinder pyramid cuboid © Classroom Secrets Limited 2018 Varied Fluency 3 Which shape is the odd one out? © Classroom Secrets Limited 2018 Varied Fluency 3 Which shape is the odd one out? The cuboid is the odd one out because the other shapes are cylinders. © Classroom Secrets Limited 2018 Varied Fluency 4 Follow the path of the cuboids to make it through the maze. Start © Classroom Secrets Limited 2018 Varied Fluency 4 Follow the path of the cuboids to make it through the maze. Start © Classroom Secrets Limited 2018 Reasoning 1 The shapes below are labelled. Spot the mistake. A. B. C. cone cuboid cube Explain your answer. © Classroom Secrets Limited 2018 Reasoning 1 The shapes below are labelled. Spot the mistake. A. B. C. cone cuboid cube Explain your answer.
    [Show full text]
  • Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2cu[Cu3o]2(PO4)4 and K2.35Cu0.825[Cu3o]2(PO4)4
    crystals Article Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2Cu[Cu3O]2(PO4)4 and K2.35Cu0.825[Cu3O]2(PO4)4 Ilya V. Kornyakov 1,2 and Sergey V. Krivovichev 1,3,* 1 Department of Crystallography, Institute of Earth Sciences, St. Petersburg State University, University Emb. 7/9, 199034 Saint-Petersburg, Russia; [email protected] 2 Laboratory of Nature-Inspired Technologies and Environmental Safety of the Arctic, Kola Science Centre, Russian Academy of Science, Fesmana 14, 184209 Apatity, Russia 3 Nanomaterials Research Center, Federal Research Center–Kola Science Center, Russian Academy of Sciences, Fersmana Str. 14, 184209 Apatity, Russia * Correspondence: [email protected] Abstract: Single crystals of two novel shchurovskyite-related compounds, K2Cu[Cu3O]2(PO4)4 (1) and K2.35Cu0.825[Cu3O]2(PO4)4 (2), were synthesized by crystallization from gaseous phase and structurally characterized using single-crystal X-ray diffraction analysis. The crystal structures of both compounds are based upon similar Cu-based layers, formed by rods of the [O2Cu6] dimers of oxocentered (OCu4) tetrahedra. The topologies of the layers show both similarities and differences from the shchurovskyite-type layers. The layers are connected in different fashions via additional Cu atoms located in the interlayer, in contrast to shchurovskyite, where the layers are linked by Ca2+ cations. The structures of the shchurovskyite family are characterized using information-based Citation: Kornyakov, I.V.; structural complexity measures, which demonstrate that the crystal structure of 1 is the simplest one, Krivovichev, S.V. Crystal Chemical whereas that of 2 is the most complex in the family.
    [Show full text]
  • A Method for Predicting Multivariate Random Loads and a Discrete Approximation of the Multidimensional Design Load Envelope
    International Forum on Aeroelasticity and Structural Dynamics IFASD 2017 25-28 June 2017 Como, Italy A METHOD FOR PREDICTING MULTIVARIATE RANDOM LOADS AND A DISCRETE APPROXIMATION OF THE MULTIDIMENSIONAL DESIGN LOAD ENVELOPE Carlo Aquilini1 and Daniele Parisse1 1Airbus Defence and Space GmbH Rechliner Str., 85077 Manching, Germany [email protected] Keywords: aeroelasticity, buffeting, combined random loads, multidimensional design load envelope, IFASD-2017 Abstract: Random loads are of greatest concern during the design phase of new aircraft. They can either result from a stationary Gaussian process, such as continuous turbulence, or from a random process, such as buffet. Buffet phenomena are especially relevant for fighters flying in the transonic regime at high angles of attack or when the turbulent, separated air flow induces fluctuating pressures on structures like wing surfaces, horizontal tail planes, vertical tail planes, airbrakes, flaps or landing gear doors. In this paper a method is presented for predicting n-dimensional combined loads in the presence of massively separated flows. This method can be used both early in the design process, when only numerical data are available, and later, when test data measurements have been performed. Moreover, the two-dimensional load envelope for combined load cases and its discretization, which are well documented in the literature, are generalized to n dimensions, n ≥ 3. 1 INTRODUCTION The main objectives of this paper are: 1. To derive a method for predicting multivariate loads (but also displacements, velocities and accelerations in terms of auto- and cross-power spectra, [1, pp. 319-320]) for aircraft, or parts of them, subjected to a random excitation.
    [Show full text]
  • 41 Three Dimensional Shapes
    41 Three Dimensional Shapes Space figures are the first shapes children perceive in their environment. In elementary school, children learn about the most basic classes of space fig- ures such as prisms, pyramids, cylinders, cones and spheres. This section concerns the definitions of these space figures and their properties. Planes and Lines in Space As a basis for studying space figures, first consider relationships among lines and planes in space. These relationships are helpful in analyzing and defining space figures. Two planes in space are either parallel as in Figure 41.1(a) or intersect as in Figure 41.1(b). Figure 41.1 The angle formed by two intersecting planes is called the dihedral angle. It is measured by measuring an angle whose sides line in the planes and are perpendicular to the line of intersection of the planes as shown in Figure 41.2. Figure 41.2 Some examples of dihedral angles and their measures are shown in Figure 41.3. 1 Figure 41.3 Two nonintersecting lines in space are parallel if they belong to a common plane. Two nonintersecting lines that do not belong to the same plane are called skew lines. If a line does not intersect a plane then it is said to be parallel to the plane. A line is said to be perpendicular to a plane at a point A if every line in the plane through A intersects the line at a right angle. Figures illustrating these terms are shown in Figure 41.4. Figure 41.4 Polyhedra To define a polyhedron, we need the terms ”simple closed surface” and ”polygonal region.” By a simple closed surface we mean any surface with- out holes and that encloses a hollow region-its interior.
    [Show full text]
  • Volume 75 (2019)
    Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information Volume 75 (2019) Supporting information for article: Lanthanide coordination polymers based on designed bifunctional 2-(2,2′:6′,2″-terpyridin-4′-yl)benzenesulfonate ligand: syntheses, structural diversity and highly tunable emission Yi-Chen Hu, Chao Bai, Huai-Ming Hu, Chuan-Ti Li, Tian-Hua Zhang and Weisheng Liu Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information, sup-1 Table S1 Continuous Shape Measures (CShMs) of the coordination geometry for Eu3+ ions in 1- Eu. Label Symmetry Shape 1-Eu EP-9 D9h Enneagon 33.439 OPY-9 C8v Octagonal pyramid 22.561 HBPY-9 D7h Heptagonal bipyramid 15.666 JTC-9 C3v Johnson triangular cupola J3 15.263 JCCU-9 C4v Capped cube J8 10.053 CCU-9 C4v Spherical-relaxed capped cube 9.010 JCSAPR-9 C4v Capped square antiprism J10 2.787 CSAPR-9 C4v Spherical capped square antiprism 1.930 JTCTPR-9 D3h Tricapped trigonal prism J51 3.621 TCTPR-9 D3h Spherical tricapped trigonal prism 2.612 JTDIC-9 C3v Tridiminished icosahedron J63 12.541 HH-9 C2v Hula-hoop 9.076 MFF-9 Cs Muffin 1.659 Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information, sup-2 Table S2 Continuous Shape Measures (CShMs) of the coordination geometry for Ln3+ ions in 2- Er, 4-Tb, and 6-Eu. Label Symmetry Shape 2-Er 4-Tb 6-Eu Er1 Er2 OP-8 D8h Octagon 31.606 31.785 32.793 31.386 HPY-8 C7v Heptagonal pyramid 23.708 24.442 23.407 23.932 HBPY-8 D6h Hexagonal bipyramid 17.013 13.083 12.757 15.881 CU-8 Oh Cube 11.278 11.664 8.749 11.848
    [Show full text]
  • Optimization of Tensegrity Lattice with Truncated Octahedral Units
    OptimizationM. Ohsaki, J. Y. of Zhang, tensegrity K. Kogiso lattice andwith J. truncated J. Rimoli octahedral units Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019 Form and Force 7 – 10 October 2019, Barcelona, Spain C. Lázaro, K.-U. Bletzinger, E. Oñate (eds.) Optimization of tensegrity lattice with truncated octahedral units Makoto OHSAKI∗a, Jingyao ZHANGa, Kosuke KOGISOa, Julian J. RIMOLIc *aDepartment of Architecture and Architectural Engineering, Kyoto University Kyoto-Daigaku Katsura, Nishikyo, Kyoto 615-8540, Japan [email protected] b School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Abstract An optimization method is presented for tensegrity lattices composed of eight truncated octahedral units. The stored strain energy under specified forced vertical displacement is maximized under constraint on the structural material volume. The nonlinear behavior of bars allowing buckling is modeled as a bi- linear elastic material. As a result of optimization, a flexible structure with degrading vertical stiffness is obtained to be applicable to an energy absorption device or a vertical isolation system. Keywords: tensegrity, optimization, strain energy, truncated octahedral unit 1. Introduction Tensegrity structures are a class of self-standing pin jointed structures consisting of thin bars (struts) and cables stiffened by applying prestresses to their members [1, 3]. Because the bars are not continuously connected, i.e., they are topologically isolated from each other, these structures are usually too flexible to be used as load-resisting components of structures in mechanical and civil engineering applications. However, by exploiting their flexibility, tensegrity structures can be utilized as devices for reducing impact forces, isolating dynamic forces, and absorbing energy, to name a few applications [5].
    [Show full text]
  • How Many Times Can the Volume of a Convex Polyhedron Be Increased By
    How many times can the volume of a convex polyhedron be increased by isometric deformations? Victor Alexandrov March 1, 2017 Abstract We prove that the answer to the question of the title is ‘as many times as you want.’ More precisely, given any constant c> 0, we construct two oblique triangular bipyramids, P and Q, such that P is convex, Q is nonconvex and intrinsically isometric to P , and vol Q > c·vol P > 0. Mathematics Subject Classification (2010): 52B10; 51M20; 52A15; 52B60; 52C25; 49Q10 Key words: Euclidean space, convex polyhedron, bipyramid, intrinsic metric, intrinsic isome- try, volume increasing deformation 1. Introduction. According to the classical theorem by A.L. Cauchy and A.D. Alexandrov, two compact boundary-free convex polyhedral surfaces in Euclidean 3-space are necessarily con- gruent as soon as they are isometric in their intrinsic metrics, see, e. g., [1]. Obviously, this is not true if at least one of the surfaces is nonconvex. In [4], the authors studied isometric immersions of polyhedral surfaces and, among other things, proved that there exists a compact boundary-free convex polyhedral surface allowing another isometric immersion which, being a nonconvex poly- hedral surface, encloses a larger volume than that enclosed by the original convex surface. This amazing existence theorem has gained new significance after the remarkable contribution of D.D. Bleecker, who explicitly built volume increasing isometric deformations of the surfaces of the Pla- tonic solids, see [3]. For example, he has shown that the surface of a regular tetrahedron can be isometrically deformed in such a way as to enlarge the enclosed volume by 37.7%.
    [Show full text]
  • Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected]
    Bridgewater State University Virtual Commons - Bridgewater State University Mathematics Faculty Publications Mathematics Department 2015 Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected] Virtual Commons Citation Burgiel, Heidi (2015). Unit Origami: Star-Building on Deltahedra. In Mathematics Faculty Publications. Paper 47. Available at: http://vc.bridgew.edu/math_fac/47 This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture Unit Origami: Star-Building on Deltahedra Heidi Burgiel Dept. of Mathematics, Bridgewater State University 24 Park Avenue, Bridgewater, MA 02325, USA [email protected] Abstract This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra. Introduction Many authors and educators have used origami, the art of paper folding, to provide concrete examples moti- vating mathematical problem solving. [3, 2] In unit origami, multiple sheets are folded and combined to form a whole; the Sonobe unit is a classic module in this art form. This workshop describes the construction and assembly of the star-building unit1, highlights a small selection of the many geometric questions motivated by this process, and introduces participants to the strictly convex deltahedra. Proficiency in geometry and origami is not required to enjoy this event. About the Unit Three Sonobe units interlock to form a pyramid with an equilateral triangular base as shown in Figure 1.
    [Show full text]